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Let $(d_1, \cdots, d_n)$ be a graphical sequence, i.e. there exists a simple graph for which this sequence is its degree sequence. I want to show the following inequality for every $k \in \{1, \cdots, n-1\}$:

$\sum_{i=1}^{k} d_i = k(k-1) + \sum_{i={k+1}}^n \min(k, d_i).$

Here is my attempt: The assumption means that there exists an adjacency matrix $A\in \{0,1\}^{n\times n}$ with $A_{ii} = 0$ such that $A^T = A$ and $\sum_j A_{ij} = d_i$.

Now we can write:

$$\sum_{i=1}^{k} d_i = \sum_{i=1}^{k} \sum_{j=1}^n A_{ij} = \sum_{i=1}^{k} \sum_{j=1}^n A_{ij} = \sum_{i=1}^{k} \sum_{j=1}^k A_{ij} + \sum_{i=k+1}^{k} \sum_{j=1}^n A_{ij} \leq k(k-1) + \sum_{i=k+1}^{k} d_i$$

The first term is $\leq k^2 - k$ because each entry $A_{ij}$ is at most $1$ while the diagonal elements are $0$. But what am I missing, why do I just get the $d_i$ instead of $\min(k, d_i)$?

Thank you in advance!

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  • $\begingroup$ I assume the $=$ is meant to be a $\leq$? $\endgroup$ Oct 12 '20 at 10:59
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I don't think using the adjacency matrix is the easiest way to think of this, as it just adds a layer of extra abstraction to deal with. Rather consider a graph $G=(V,E)$ realising this sequence, and look at the set $S = \{v_1, v_2, \dots, v_k\}$ of vertices in $G$ such that $deg(v_i) = d_i$.

Click the spoiler below for an outline of a proof:

Each of the $k$ vertex in $S$ has at most $(k-1)$ neighbors in $S$, and so contributes at most $(k-1)$ to the sum $\sum_{i=1}^{k}d_i$. Further, each vertex $v_j$ of $V-S$ is adjacent to at most $\min\{k, deg(v_j)\}$ vertices of $S$, and so contributes at most $\min\{k, deg(v_j)\}$ to the sum $\sum_{i=1}^{k}d_i$

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